VNU. JOURNAL OF SCIENCE, Mathematics - Physics. T.xx. N03AP. 2004
P R IN C IP A L CO M PONENT A N A LYSIS FO R F IE L D SEPARA TIO N
A b s t r a c t . T h e a r tic le c o n ta in s d iffe re n t te c h n iq u e s o f g e o p h y s ic a l da ta p ro c e s s in g b y u s in g s o ftw a re M a th e m a tic a . F ie ld s e p a ra tio n is o n e o f th e m o s t im p o rta n t p ro b le m s in g e o p h y s ic a l d a ta in te rp re ta tio n . F o r p o te n tia l fie ld s , w h e n th e re a re o b s e rv a tio n a l da ta fo r th e b o th th e p ro file s u rv e y a n d a re a s u rv e y . T h e fie ld s e p a ra tio n b e c o m e s a p ro c e s s o f e s tim a tio n o f low fre q u e n c y c o m p o n e n t, i.e ., th e re g io n a l a n o m a ly , o n th e o n e h a n d , a n d hig h- fre q u e n c y fie ld c o m p o n e n t, i.e . th e re s id u a l o r lo c a l a n o m a ly o n th e o th e r ha n d .
P rin c ip a l c o m p o n e n ts a n a ly s is fo r fie ld s e p a ra tio n p ro v id e s im m e d ia te in s ig h t in to th e s tru c tu re o f fie ld d a ta a n d is a p p lie d fo r m o d e lin g th e a n o m a ly fie ld th a t c o n ta in s d iffe re n t b o d ie s .
T h e c a lc u la tio n p ro c e s s is re a liz e d b y u s in g th e c o m p u te r a lg e b ra ic sys te m (m a th e m a tic a ). T h e re s u lt o f re s e a rc h is u s e d in g e o p h y s ic a l fie ld s e p a ra tio n fo r g e o p h y s ic a l in te rp re ta tio n .
1. I n t r o d u c t io n
T r a d it io n a l in te rp re ta tio n s o f th e geophysical d a ta ha ve cocentrated on one o r two preselected va ria b le s o r fu nction s o f the variables. How ever, the m u ltiv a ria te s tru ctu re o f the d a ta suggests th a t s ta tis tic a l techn iqu es o f m u ltiv a ria te an a lysis are appropriate.
P r in c ip a l com ponents a n a lysis as a m u ltiv a ria te e xp lora tory techniques provides a u sefu l s ta rtin g p o in t fo r fu r th e r investigations. It m ay also p ro vid e in s ig h t in to the geological processes u n d e rly in g th e data. It is a m ethod fo r decom posing the tota l v a ria tio n o f m u ltiv a ria te observations in to lin e a r ly independent com ponents o f decreasing inpotance.
In th is a rticle , the p r in c ip a l com ponent ana lysis is u se d fo r fie ld sep a ra tion and in te rg ra te d d a ta processing.
1. A p p l i c a t io n o f t h e p r i n c i p a l c o m p o n e n t a n a ly s ic : F i e l d s e p a r a t io n f o r a r e a l s u r v e y d a ta
C o n s id e r th e a p p lica tio n o f th e p rin c ip a l com ponent a n a lysis fo r fie ld separation w hen there are a re a l survey data. L e t the set o f ran dom va lu e s X ] ,...,XN be presented by tw o-d im en sion al d a ta f ile (area l surv ey data) fo r the sam e p h y sic a l field, in the form o f m a trix o f N row s a n d n colum ns. T h e a lg orith m o f fie ld sep a ra tion in clu d e the follow ing operation:
C alculation o f the mean for each profile:
T o n T i c h A i
D epartm ent o f Physics, College o f Science, V N U
w here x kj are the observed fie ld d a ta fo r th e k th p o in t o f the ith profile.
1
2
Ton Tich Ai
C alculation o f covariance for each p a ir o f profiles
bjj = - £ ( X k i - x i)(xkj ~ x j) . i. j = 1.2, -
Construction o f covariance m atrix B:
fill b|2 '»1
I bN2 ... I
w here bii i s the observ ed d a ta v a ria n ce fo r the ith p ro file a n d 6y = bjị C alculation o f the m a x im u m eigenvalue /.max by solving the m a trix equation.
' b u -X . bl2 ... b]N b?',! - baN
. .1 b N2
T h e eigenvalues o f th is e q uation ?.J, ...,XN a re the roots o f the equation, w ith the d e te rm in a n t o f (B - I) b e in g e q ua l to zero. Father, it is n ecessary to select the m a xim um eigen valu e am ong the o bta in ed roots.
O btaining the m a x im u m eigenvector o f m atrix B, w h ic h correspon d to the Ảmux , w ith the a id o f th e set o f equations:
(bj| - ^max)®11 + b ia®12 +... + b |f b12a.
+ b ,2a 12+... + b lN a 1N = 1 + (b22 -? - mils)a,2 + ... + b2Na 1N =
b lN a l l + ^2Na 12 + ... + (^NN - ^max )®1N - 0
T h e eigenvector a ](a11ta i2>-»>&iN) is de te rm ine d in term s o f n orm a liza tion
= 1 , i =1, 2, T h e p h y sica l sense of such n o rm a liz a tio n im p lie s th e e xpression of the tra nsfo rm ed d a ta at the sam e scale as the p rim a ry fie ld data.
F inding o f the first p rin cip a l component Y ] = a 'X o r
= (*11.*12
W e ca n re g a rd th e v a lu e s Y ) K (k = 1,2, ...,n) as the w e ig h t coe fficien ts fo r each point o f the fie ld data. In th is connection, the valu es a ! j (i = 1 , 2 , N) d e te rm in e th e w eight coefficient for each profile.
E stim ation o f the field component, cha ra cteriz ed by m a xim u m varia nce, u sin g the m a trix expression:
P r i n c i p a l c o m p o n e n ts a n a ly s is for...
y i i ® n + X 1 y n a12 + x2 . • y u « iN + XN .. rt'g _
ki ~ Y,2
(a n , a 12, ... a 1N) + X i =y I2a i 1 + X i y12a12 + x2 .. y I2a lN + X N Y ,„ >
. y in a l l + X 1 y i na 12 + x2 • y i na iN + * N ,
T h e fie ld com pon ent ha vin g the m a xim um varia nce, e nsures the e stim a tio n o f the regional an om a ly w h e n X max = ( 7 0 - 9 0 % ) ^ A., . S ince x ỹ g is the e stim a tio n o f the regional anom aly, th e n the d iffe re nce xjjj0 = Xjji - x£jg w ill be the e stim a tio n o f the loca l one.
O n b asic o f th e presented ab ou t a lg orirhm , the p rogram fo r c a lc u la tin g regional and local an om a lie s o f p o te n tia l fie ld is m ade by a u th o r in lan gu ag e “M a th e m a tica ”:
« S ta tis tic s 'D e s c rip tiv e S ta tis tic s ' n = D im e nsion s[d ata O ];
n1 = n Ị ỊIỊỊ : n 2 = n [[2 ; b = Id e n tity M atrix(n2];
D o [x [i] = M ea n[d ataO [[i]]], (i, n1}]
D o (D o [b [[i, j j] = Sum [(dataO [[k, i]] -
X [i])(da ta 0([k, j]] - x [j]) , {k . n1>]/n1. {». 1. n1 }]. {j, 1. n1}]
d = E ig en vecto rs(b );
{d([1 ]].d a ta 0 };
d a ta i = T ra n sp o se [% ].{d |[1 ]]};
Do[Do[data1([i,j]]= data1[[i. j]] + x[i], 0, n1}J, {i, n2}J
dto = L istC o nto urP lo tfd ata O , C o n to u rS h a d in g -> F alse , C o n to u rs -> 20.
F ra m e L a b e l -> {"x .1 0 0 m ", "y.1 0 0 m "}, C o n to u rS ty le -> RGBColorJO, 0, 1]J;
dt1 = L is tC o n to u rP lo t[
d a t a i, C o n to u rS h a d in g -> False , C o n to u rs -> 20, F ra m e L a b e l -> {"x .1 0 0 m ",
"y .1 0 0 m "}, C o n to u rS tyle -> R G B C o lo r[0, 0 , 1]J;
d t2 = L istC o nto urP lo tỊd ata O - d a ta i, C o n to u rS h a d in g -> F a ls e , C o n to u rs ->
4 0 , F ra m e L a b e l -> {"x .1 0 0 m '\ "y-1 00 m "}, C o n to u rS ty le -> R G B C olo rỊO , 0, 1J];
M o d e l i n g d i f f e r e n t f i e l d s e p a r a t io n s . T o d em onstrate the fie ld separation a b ility o f the m ethod, in th is article, the m odel o f three spheres o f d iffe re n t pra m e te rs is selected..
T h e re su lts o f c a lcu la tio n are p resented in fig u res 1, 2, 3.
F ig . 1 Total Anomalies F 'g-2 Regional Anomalies F ig.3 Local Anomalies
Ton Tich Ai
B y u sin g p r in c ip a l com ponents ana lysis w e m ay e m phasize d iffe re n t com ponents from total an om a lie s in d ependence o f o u r in te rp re ta tion goal. T h e m ethod wa s sim p lifie d to enable e a sie r a n d th u s p ossibly g eological in terp retation o f g eoph ysical d a ta in d iffe re nt conditions.
R e fe r e n c e s
1. Ton T ich A i, Mathematica for engineer, N ational U niversity Publisher, H an oi 2003.
2. To n T ic h A i, Applied Geophysics .U niversity M in is try Publisher, H anoi 1988.
3. Stephen Wolfram , Mathematica. A ddison-W esley Publishing Company, Inc. 1988.
4. Tafcev.G.P, Sokolov K.P., Geological interpretation o f magnetic anomalies. Neilra.
Leningrad. 1981.
5. N ik itin A . A., Statistical Processing o f Geophysical Data. Electrom agnetic. Research Center.
Moscow 1993.
6. J.'IYoehimczyk and F.Chayes., Some Properties o f Principal Component Scores. Mathematiacal geology, Vol. 10, NO. 1,1978
2. C o n c lu s io n s
